I usually use github repositories or repositories local to any organization that employs me. Now I should install a local standalone repository with git for a team. I could get it to work with gitweb + standalone git server but my team leader said that there could be a problem because it only allowed "one user". But I could set up 2 different ssh keys and commit from 2 different machines as 2 different users.

Note that this site is not about "is this software fitting my needs" – but about "what software meets these requirements". So I've interpreted your question as specifying the requirements, and answered it accordingly. Hope it fits!
– Izzy♦Aug 12 '17 at 23:51